

where \(\hat{g}\)  is the double-valued representation of \(g\) , \(U^n_{g(j)g(i)}=\sum _m R_{nm}(g) W_j(g) U^m_{ji} W^\dagger _i(g)\) , \(U_{g(j)g(i)}=W_j(g) U_{ji} W^\dagger _i(g)\)  and \(R(g)\)  is a vector representation of \(g\) . The new symmetry operations, each of which involves a symmetry operation \(g\)  followed by a gauge transformation \(W_i(g)\) , form a larger group that is known as the projective symmetry group (PSG)[1]}, [2]}.
